{"id":6530,"date":"2021-09-27T14:38:31","date_gmt":"2021-09-27T09:08:31","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=5706"},"modified":"2025-11-09T23:01:19","modified_gmt":"2025-11-09T17:31:19","slug":"what-is-a-comparison-test","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-a-comparison-test\/","title":{"rendered":"What is a comparison test?"},"content":{"rendered":"\n

Overview<\/strong><\/h2>\n\n\n\n

Comparison test<\/a>, which is also known as limit comparison test. we use this method to test the convergence of an infinite series.<\/p>\n\n\n\n

Convergent sequence<\/strong>– A sequence Sn is said to be convergent when it tends to a finite limit. That means the limit of a sequence Sn will be always finite in case of convergent sequence.<\/p>\n\n\n\n

Divergent sequence<\/strong>– when a sequence tends to \u00b1\u221e then it is called divergent sequence.<\/p>\n\n\n\n

Oscillatory sequence- <\/strong> when a sequence neither converges nor diverges then it is called oscillatory sequence.<\/p>\n\n\n\n

Note<\/strong>– a sequence which neither converges nor diverges , is called oscillatory sequence.<\/p>\n\n\n\n

A sequence converges to zero Is called null.<\/p>\n\n\n\n

Series<\/strong><\/h2>\n\n\n\n

Infinite series-<\/a> <\/strong>If  is a sequence , then is called the infinite series.<\/a><\/p>\n\n\n\n

It is denoted by  <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Examples of infinite series-<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Covergent series – <\/strong> suppose n\u2192\u221e , Sn\u2192 a finite limit  \u2018s\u2019  , then the seiresSn is said to be convergent .<\/p>\n\n\n\n

We can denote it as,<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Divergent series<\/strong>–  when Sn tends to infinity then the series is said to be divergent.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Oscillatory series- <\/strong> whenSn does not tends to a unique limit (finite or infinite) , then it is called Oscillatory series.<\/p>\n\n\n\n

Properties of infinite series <\/h2>\n\n\n\n

1. the convergence and divergence of an infinite series is unchanged addition od deletion of a finite number of term from it.<\/p>\n\n\n\n

2. if positive terms of convergent series change their sign , then the series will be convergent.<\/p>\n\n\n\n

3. let   converges to s , let k be a non-zero fixed number  then converges to ks.<\/p>\n\n\n\n

4. let converges to \u2018l\u2019 and  converges to \u2018m\u2019.<\/p>\n\n\n\n

Comparison test<\/a><\/strong><\/h2>\n\n\n\n

Statement- Suppose we have two series of positive terms  <\/strong><\/strong>then, <\/strong><\/strong> , where k is a finite number , then both series converges or diverges together.<\/strong><\/p>\n\n\n\n

Proof-<\/strong><\/p>\n\n\n\n

we know that by the definition of limits, there exist a positive number epsilon(\u03b5)<\/p>\n\n\n\n

Which is very small. Such that<\/p>\n\n\n\n

According to definition( comparison test)<\/p>\n\n\n\n

for n>m  , that means<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

ignoring the first m terms of the series,<\/p>\n\n\n\n

we get<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

there will be two cases-<\/p>\n\n\n\n

case-1: is convergent , then<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

where r is finite number<\/p>\n\n\n\n

From (1), <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n
\"\"\/<\/figure>\n\n\n\n

Therefore  \u2211u_n  is also convergent.<\/p>\n\n\n\n

Case-2: :\u2211v_n  is divergent, then<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

From eq. (1)<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Then<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

From (2)<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Hence, is also divergent.<\/p>\n\n\n\n

Example: Test the convergence of the following series.<\/strong><\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Sol. We have  <\/p>\n\n\n\n

First we will find un  and vn the <\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n
\"\"\/<\/figure>\n\n\n\n

And<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Here, we can see that, the limit is finite and not zero,<\/p>\n\n\n\n

Therefore, and converges or diverges together.<\/p>\n\n\n\n

Since is of the form   where p = 2>1<\/p>\n\n\n\n

So that , we can say that,<\/p>\n\n\n\n

is convergent , so that will also be convergent.<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n